In 1956, a 25-year-old mathematician named John Milnor was investigating mathematical objects in seven dimensions. He wasn't trying to make a revolutionary discovery. He was studying a family of geometric spaces using two different mathematical approaches. Then something deeply unsettling happened. One argument suggested that these spaces were spheres. Another seemed to prove that they could not be spheres. Both arguments appeared mathematically correct. Milnor was baffled. Years later, in an interview, he recalled: “It was a complete accident, and certainly startled me.” The apparent contradiction forced him to question an assumption that mathematicians had taken for granted. What if two spaces could be topologically identical but have fundamentally different smooth structures? That was the answer. Milnor had discovered what became known as exotic spheres: seven-dimensional spaces that are topologically equivalent to ordinary seven-dimensional spheres but not smoothly equivalent. His seven-page research paper, On Manifolds Homeomorphic to the 7-Sphere, appeared in the Annals of Mathematics in September 1956. The discovery transformed differential topology. In later work with Michel Kervaire, Milnor established that the seven-dimensional sphere admits exactly 28 distinct oriented smooth structures, up to orientation-preserving diffeomorphism. In 1962, Milnor received the Fields Medal. In 2011, he received the Abel Prize for his pioneering discoveries across topology, geometry and algebra.

Saturday's puzzle

Well done to Senan Keena who won the ‘Guess the Number’ Maths Week Challenge!

A doctor drove around America in a van with an ice pick and a hammer. He would put a patient under with an electric shock, lift the eyelid, and drive the pick in through the top of the eye socket. Walter Freeman did about 3,500 of these. He kept no meaningful notes. The transorbital lobotomy. Freeman developed it in the 1940s so that the operation could be done without a surgeon, without an operating theatre and without anaesthesia, in about ten minutes. He travelled in a vehicle he called the lobotomobile and performed them in state hospitals across the country, sometimes a dozen or more in a day, sometimes in front of an audience. He was not a surgeon. He had no surgical training. He operated on about 3,500 people. At least nineteen of them were children; the youngest was four years old. Now the part that belongs on this account. The transorbital lobotomy was performed overwhelmingly on the people least able to refuse it — patients in underfunded, overcrowded public institutions, where the appeal of a ten-minute procedure that made a difficult patient quiet was administrative rather than medical. In the segregated American South and in segregated wards elsewhere, that population was disproportionately Black. Black patients in state institutions were more likely to be committed on thin grounds in the first place — you have had Crownsville in this feed, where a man was institutionalised for startling a horse — and once inside, they were in exactly the pool Freeman was operating on. The documentation of who specifically received these operations is poor, which is itself part of the story: the records of what was done to people in segregated state hospitals are patchy because nobody was required to keep them well. He kept operating into the 1960s. He was finally stopped in 1967 when a patient he was operating on for the third time died of a brain haemorrhage. A man with an ice pick, a van, and no surgical qualification, working his way through the wards of American public hospitals for twenty years. These are the histories that must not disappear. Get our book and uncover the painful stories too often left out of the record. savvypurchase.samcart.com/pr…

Friday's puzzle

Great excitement this morning as our Guess the Sweets competition winners were celebrated @mathsweek @NsKillea @templemoretown.

⭐️ Maths Week Ireland Staff Puzzle Winners ⭐️ Congratulations to all the secret Mathletes hiding in plain sight in St Catherine’s College. Mrs Nixon was our fifth winner but she was camera shy! 🥳🧩🧮 #mathletes #staffpuzzlewinners #notjustmathsteacherslovemathspuzzles

Wise heads at St Catherine’s

Great work

Two MIT mathematicians have proposed a significant change to the future of mathematical research. In an essay published on 21 October, Pavel Etingof and Slava Gerovitch argue that mathematicians should increasingly be evaluated for their understanding, judgment, and creativity rather than simply for being the first to prove a theorem. They suggest that future PhD theses need not necessarily contain the first proof of a result. Originality could instead come from new ideas, connections, explanations, or methods. They also propose greater emphasis on oral examinations, collaborative research, and AI-assisted mathematical exploration. Their central argument challenges a longstanding academic tradition: mathematical achievement should not be measured primarily by who proves something first.

AI is changing mathematics! Where does the conversation go from here? The American Mathematical Society (AMS) has launched AI and Mathematics: Selected Resources, a curated collection to help mathematicians explore the opportunities, questions, and evolving discussions surrounding AI and mathematical discovery. The web page brings together resources on AI-assisted research, proof verification, mathematics education, emerging tools, and expert perspectives with opportunities to explore new ideas, learn from others, and engage with the evolving role of AI in mathematics. The AMS is committed to connecting our community and fostering thoughtful, informed conversations as the impact of AI on mathematics continues to unfold. Have a resource to recommend? Email: ai@ams.org Explore resources. Link in comments.

Why are reading scores falling in many countries? PISA points to rising levels of "hasty reading", where students answer quickly but incorrectly, and growing difficulties with evaluating information, critical thinking and using multiple sources. oecd.org/en/publications/202…

🧮🎉 Tá Seachtain an Mhata linn! 🎉🧮 🌟 Faigh amach faoi na hacmhainní Gaeilge a thacaíonn le teagasc agus foghlaim an Mhata ar #TairseachCOGG. 📚🔢 Acmhainní do gach leibhéal 💡 Gníomhaíochtaí spraíúla 🌐 tairseachcogg.pulse.ly/uvagu… #MathsWeek #SeachtainanMhata

📐➕ Looks like we’ve solved for the students’ FAVOURITE Maths teacher! 🤩📊 This Maths Week, the fun with Mr Culloty is adding up, the smiles are multiplying, and the good vibes are off the charts! ➗✖️ Who knew Maths could be this much fun? 💙 #MathsWeek #MathsIsFun

Lots of active learning on this bright October morning @NsKillea @templemoretown @mathsweek

🔢📐 MATHS WEEK! ➗🎉 As part of Maths Week, we enjoyed a range of fun activities throughout the week in Meán Scoil Mhuire! Congratulations to our winners, table 15 and runners up table 14 in the 1st year Maths Quiz.

Nothing beats solving a great puzzle @NsKillea @mathsweek

➗😂 Maths Week couldn’t have come at a better time! The sums aren’t adding up, the equations are causing confusion, and even the calculators are giving up on the kids! 🧮🤯 Time to multiply the fun and divide the stress! 📐✖️ #MathsWeek #MathsIsFun

What an effort! What an achievement. Pushed all the way by @CaheraghNS. Top of the weekly leaderboard for the @mathsweek targetboards. All Ireland champions has a nice ring to it!

RT @ats: 世界最高峰の数学者テレンス・タオが、いま「数学をする時間」を削ってまでAIへの対応に追われている。問題はAIが数学を解けることではない。数学という営みそのものが、何を価値とするのかを問い直されていることだ。 UCLAの数学者でフィールズ賞受賞者のTerenc…